Upgrade from 8.2.1 to 8.4.4. Page not found

I am using 8.2.1, at the Welcome dashboard (/index.php/dashboard/welcome) I see:
The latest version of concrete5 is 8.4.4. You are currently running concrete5 version 8.2.1.
So, I click Update, then I get:
Page Not Found
/index.php/dashboard/system/backup/update this is the link not found.
If in System and Settings I click Apply Update and I get. No updates are ready to be installed.

Yes, that is what I want. But the original install was 7.5.13, then I did the online upgrade to 8.2.1. So if I rewrite everything in the root (public_html) folder, wouldn't that mess things up? I just want to be certain nothing bad will happen if I do this. I will back things up, but it is still a mess if it fails. Or did I misunderstand what I was supposed to do?

You did misunderstand..
You place the unzipped 8.4.4 package in the ROOT/UPDATES folder so that when you visit the system & settings page you will find the update to 8.4.4 waiting to be activated.
So just to reiterate this, you do not put the 8.4.4 package in your 'root' folder, you put it into your 'root/updates' folder..
In fact, if you have access to the server (cPanel or similar) you should put the zipped 8.4.4 file into the 'root/updates' folder and then extract it..